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We consider a stationary continuous model of random size population withnon-neutral mutations using a continuous state branching process withnon-homogeneous immigration. We assume the type (or mutation) of the immigrantsis random given by a constant mutation rate measure. We determine somegenealogical properties of this process such as: distribution of the time tothe most recent common ancestor (MRCA), bottleneck effect at the time to theMRCA (which might be drastic for some mutation rate measures), favorable typefor the MRCA, asymptotics of the number of ancestors.

A population model with non-neutral mutations using branching processes with immigration